Embedding the bicyclic semigroup into countably compact topological semigroups

TL;DR

This paper investigates the embedding of the bicyclic semigroup into various topological semigroups, establishing conditions under which such embeddings are impossible or possible, and providing explicit examples.

Contribution

It proves that pseudocompact squares exclude dense copies of the bicyclic semigroup and constructs a consistent example of a pseudocompact semigroup containing it.

Findings

Pseudocompact squares do not contain dense copies of the bicyclic semigroup.

A consistent example of a pseudocompact Tychonov semigroup containing the bicyclic semigroup is constructed.

The study links algebraic properties of the bicyclic semigroup with topological compactness conditions.

Abstract

We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonov semigroup containing a copy of C(p,q).